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@daniel_p i think there should be at least one scientific description of what we they are doing here, it would be a pleasure
for me to collect all the informations we have here, but it would take at least 2 or 3 monthes ( hello bachelor thesis, actually i am trying to find someone who would find that interesting )

cheers 

I try to. But navigating using the parameters variations instead of "natural" 2D zooming in UF is quite tough.

I know there is nothing revolutionnary in the images I post in this thread compared to all the ground-breaking innovations we can see here, but it's just to show that zooming into 3D fractals that look like whipped cream at first sight can reveal interesting spirals and so on...

Just a note to anyone reading the thread who may be unfamilier with |z| as used in standard maths notation and/or as used in Fractint and Ultra Fractal.

If z is complex as (x+iy) then in standard maths notation |z| = sqrt(x^2 + y^2)

BUT in Ultra Fractal and Fractint then |z| = x^2 + y^2 and cabs(z) is used as sqrt(x^2 + y^2) (cabs for complex absolute).

I was prompted to mention this after IQ's query regarding the correct scaling of the DE value, I also couldn't work out how "0.5" was arrived at. In UF and Fractint:

0.5*log(|z|) is equivalent to log(|z|) in standard math notation

And at one point I considered that maybe the 0.5 came from mistranslations between fractal program and standard math notations.

But I don't think the 0.5 comes from that, I think it really comes from the fact that the value calculated is only an *estimate* and the 0.5 is simply used to avoid cases where the estimate is larger than the true distance - but that's just my opinion
In fact the DE value I use for the Normals is assumed to be correct without the "0.5", I do however multiply the DE value by 0.5 before using the value as a step distance during ray stepping.
Also I should mention that (based on how I had to scale the DE values used to compute the normals to get correct lighting) I found (experimentally) that the visibly correct (Ultra Fractal) final DE calculation was:

            dist = 0.5*sqrt(@mpwr-1.0)*log(magn)*sqrt(magn/|dzri|)

for a Julibrot where @mpwr is the power in z^power+c, magn is |final z| (i.e. x^2+y^2) and dzri is the final derivative value.
However looking at IQ's derivation I suspect that:

            dist = 0.25*@mpwr*log(magn)*sqrt(magn/|dzri|)

May be more accurate Edit: Just tried it and it doesn't seem to be better, in fact it definitely overestimates at higher powers - I not only got gradually darker and darker shading but started getting gross overstepping as well when the method using sqrt(@mpwr-1) still worked fine.
Just to add I actually checked it on quaternions and the White/Nylander.

I'm not sure what went wrong with the z^3+c when I tried it (or the other higher powers where you use abs and sign) maybe I'm missing something obvious.
Anyway the formula I quoted here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8475/#msg8475 can be expanded to use reals for a given power and then optimised - that formula definitely exactly matches the trig version (at least for integer powers from 2 to 60, I haven't checked fractional or negative powers).

Here is a very simple general procedure to obtain sin(p.a) and cos(p.a) if you know the value for sin(a) and cos(a) :

Let S=sin(a) and C=cos(a) and make a complex number z=S+iC, and put z1=z

Do z1=z1.z,   now real(z1)=cos(2a) and imag(z1)=sin(2a)  (z1=C^2-S^2 +i.2.S.C)

Do again z1=z1.z now real(z1)=cos(3a) and imag(z1)=sin(3a)

(because (sin(2a)+i.cos(2a))(sin(a)+i.cos(a) gives as real part sin(2a)cos(a)+cos(2a)sin(a)=sin(3a) and as imaginary part cos(2a)cos(a)-sin(2a)sin(a)=cos(3a) )

...
..(the (p-1)-th time)  real(z1)=cos(pa) and imag(z1)=sin(pa)

So no need for very long complicated formulas if you want do to a non-trig p-th power...
The drawback is that you need sin(a) and cos(a) which involves a square root.

For even powers this can be adapted so that you only need (sin(a))^2 and (cos(a))^2, which does away with the square root. However, when using a distance estimate based on the derivative, one always needs also the value for sin((p-1)a) and cos((p-1)a), so getting a value for sin(a) and cos(a) cannot be avoided then.

Also, if you want to do powers<1, it is no good of course.

Hi ! Thank you for sharing this.
I'm playing with incendia (which is awesome) since yesterday.

And with UF5 (demo ) and your formula.
Sadly, not good enough to understand the math behind (*sigh*) i just click here and there and found that i could do some interestring rendering with the "Truly 3D Mandelbrot". i don't know how "true" is my rendering and how it could be even close to anything related to a mandelbrot fractal but... it's nice.

I've seen the pic of "3D Mandelbrot" seen from the inside... it's truely awesome. But it look like most of 3D Mandelbrot rendering are done with custom renderer and not distributed at all... am i correct ?

Thx again !!

PS about the image below. "Truely 3D mandelbrot" after randomly playing with parameters

Edit : i finally found how to adjust parameters to have a 3D Mandelbrot, adjust accuracy and detail.
It is, mostly, all about "Solid Thresold" to be able to dive into details

On the question which factor should precede the analytical distance estimate formula :

is it or ?

Let's change notation first : for the spherical case for any power I use DE=log(R)*sqrt(R)/(dR)/2, where dR is the derivative. So my factor is just 0.5.

To test how accurate this is, I took a Julia with seed (0,0,0) which produces a sphere of radius 1, so we now know the real distance for any point on a ray.

I'm printing the results below for a point progressing from a distance 2 from the surface of the sphere. The first column is the number of iterations, the second is the calculated DE, and the third is the real distance. As the point is still "far" away, and the number of iterations to reach bailout is low, then DE is exaggerated. (that's why an extra factor to temper DE is always needed). However, as the point gets closer, DE becomes very accurate.
The numbers are from a power 8 Julia, but I see almost exactly the same numbers for a degree 2.

power=8
start
         DE                        real distance
1 , 3.29583686625158 , 1.99999999999248
1 , 0.407849058786746 , 0.352081566866694
2 , 0.158627173103063 , 0.148157037473318
2 , 0.0711605799066145 , 0.0688434509217899
2 , 0.033810346387068 , 0.033263160968481
3 , 0.01649105643562 , 0.0163579877749456
3 , 0.00814527732540994 , 0.00811245955713247
3 , 0.00404797039654616 , 0.00403982089442678
4 , 0.00201786652103339 , 0.00201583569615593
4 , 0.00100740958383335 , 0.00100690243564117
4 , 0.000503324618422875 , 0.000503197643723752
5 , 0.000251567358870418 , 0.000251535334513164
5 , 0.000125759953482483 , 0.000125751655076556
5 , 6.28740467193265E-005 , 6.28716783381833E-005

hooo fluffy \o/